In the poem, "The Ballad of Father Gilligan", why was the priest weary day and night?

Father Gilligan in the ballad by Yeats, "The Ballad of
Father Gilligan," is tired because his people are dying off so fast.  His job is to go
to them, comfort them, and give them last rites when they are near death.  And they are
dying so fast that he doesn't get any rest.

Here are the
lines that are relevant to your question:

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Once, while he nodded on a chair,
At
the moth-hour of eve,
Another poor man sent for him,
And he began to
grieve.

"I have no rest, nor joy, nor peace,
For people die
and die';
And after cried he, "God forgive!
My body spake, not
I!'

He knelt, and leaning on the chair
He prayed and fell
asleep;
And the moth-hour went from the fields,
And stars began to
peep.

Notice, "Another" man
sent to him.  Another man is dying.  And the priest responds by beginning to grieve. 
The following lines reveal, though, that he is grieving for himself, rather than for the
man who is in the process of dying.  He grieves, and he is tired, because he just can't
keep up. 

He is so tired that even though he is horrified
at what he says--"I have no rest, nor joy, nor peace,/For people die and die"--he falls
asleep while praying and leaning on a
chair.